The GSV Index

Abstract

One of the basic properties of the local Poincaré – Hopf index is stability under perturbations. In other words, if a vector field has an isolated singularity on an open set in ℝⁿ and if we perturb it slightly, then the singularity may split into several singular points, with the property that the sum of the indices of the perturbed vector field at these singular points equals the index of the original vector field at its singularity. If we now consider an analytic variety V defined by a holomorphic function ${\rm f :} (\mathcal{C}^{{\rm n + 1}} {\rm , 0) } \to {\rm }(\mathcal{C}, 0)$ with an isolated critical point at 0, and if v is a vector field on V, with an isolated singularity at 0, then one may like “the index” of v at 0 to be stable under small perturbations of both, the function f and the vector field v. This leads naturally to another concept of index, called the GSV index, introduced by X. Góomez-Mont, J. Seade and A. Verjovsky in [71, 144] for hypersurface germs, and extended in [149] to complete intersections. In this chapter we define this index and we study some of its basic properties. We first do it when the ambient space is an isolated complete intersection singularity (ICIS for short), then we explain the recent generalization in [34] to the case where the ambient variety has nonisolated singularities; this relies on a proportionality theorem similar to the one proved in [33] for the local Euler obstruction, that is discussed later in the text.

In the following chapters we will study other related indices: the GSV index can be interpreted via Chern.Weil theory as the virtual index introduced by D. Lehmann, M. Soares and T. Suwa in [111], that we study in Chap. 5. And if the vector field v is holomorphic, then the GSV index also coincides with the homological index of Góomez-Mont [68], that we describe in Chap. 7. There is also a recently defined logarithmic index in [7], which coincides with the homological index and therefore, for ICIS, with the GSV index.